Many cities and counties are fighting the dog bite problem by enacting so-called "leash laws" and "dangerous dog laws." (Read the article about one such city.)
A leash law requires dogs to be under control by a leash and a competent handler. A dangerous dog law goes further, setting forth what conduct of dog and owner shall be illegal (including the dog being off leash, trespassing or at large), establishing consequences for dog and owner, and providing reasonable rights of notice and hearing before the imposition of the more severe consequences.
Every city, county and state should have a dangerous dog law. Overlapping laws are not necessary, however. In other words, a single, state law could suffice if meant to cover all cities and counties. If there is no state law, a county law is the next best alternative; and if the state and county have no such law, a city could pass its own.
This applies to your state, county and city. If you don't have such a law, or want ideas for a well-balanced law, see the Model Dangerous Dog Law on Dog Bite Law. To deal with the bad owners themselves, every jurisdiction also should have an Irresponsible Dog Owner Law.